Little research has been done on these units. Confederate units were paid by the Central government whereas most of the units were paid by their respective state. You will find men from various states and many men who were formerly in the U.S. Army as officers as well as partisan ranger units which were consolidated to make up the various Cavalry Units.
All "regular" units and the vast majority of officers actually served in the "Provisional Army of the Confederate States" (PACS) and not the "Army of the Confederate States of America" (ACSA). Only a handful of high ranking officers were appointed to the ACSA to ensure that they would always outrank all militia, state, and PACS officers.
"The Provisional Army of the Confederate States (PACS) was authorized by the Provisional Confederate Congress on February 28, 1861, and began organizing on April 27. Virtually all regular, volunteer, and conscripted men preferred to enter this organization since officers could achieve a higher rank in the Provisional Army than they could in the Regular Army. If the war had ended successfully for them, the Confederates intended that the PACS would be disbanded, leaving only the ACSA.
The Army of the Confederate States of America (ACSA) was the regular army, provided for by Act of Confederate Congress on March 6, 1861. It was authorized to include 15,015 men, including 744 officers, but this level was never achieved. The men serving in the highest rank as Confederate States Generals, such as Samuel Cooper and Robert E. Lee, were enrolled in the ACSA to ensure that they outranked all militia officers. ACSA ultimately existed only on paper. The organization of the ACSA did not proceed beyond the appointment and confirmation of some officers. Three state regiments were later denominated "Confederate" regiments but this appears to have had no practical effect on the organization of a regular Confederate army and no real effect on the regiments themselves."
The Confederate Congress, on March 6, 1861, passed 'an act to provide for the public defense'. It called for the estrablishment of a regualr standing army, which would have the same organization as all state and volunteer units. The army was to consist of 6,000-10,000 men. I have been able to identify nine regiments (numbered 1st through 9th with no rosters for the 6th, 7th, and 8th) labeled as 'Confedrate' instead of any state name (i.e. 1st Confederate, etc.). The 5th Confederate may have been an Engineer unit, but the roster contains only two names.
The infantry regiments were to have a prescribed strength of 1145 according to the regulations established on March 6, 1861. Five Regimental Officers, and 10 companies each were to consistof 14 Command personnel and musicians, and 64-100 privates. The number of privates was increased to a minimum of 125 per company per the new regulations of October 11, 1862, however this number was never achieved.
The Cavalry Unit numbers run through 20, although some numbers are missing. It is unknown whether the National Archives does not have rosters or the units did not exist.
The Cavalry regiments were established with a strength of 755 on March 6, 1861. 5 regimental officers, and 10 companies consisting of 16 Headquarters personnel and 60 privates (increased to 80 privates at a minimum on October 11, 1862… again, never reached).
The National Archives index contains 80,552 names in the Confederate Regulars.
The Museum of the Confederacy has an inventory of their resources on these units.(PDF format).